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 A156547 Decimal expansion of the central angle of a regular dodecahedron. 2
 7, 2, 9, 7, 2, 7, 6, 5, 6, 2, 2, 6, 9, 6, 6, 3, 6, 3, 4, 5, 4, 7, 9, 6, 6, 5, 9, 8, 1, 3, 3, 2, 0, 6, 9, 5, 3, 9, 6, 5, 0, 5, 9, 1, 4, 0, 4, 7, 7, 1, 3, 6, 9, 0, 7, 0, 8, 9, 4, 9, 4, 9, 1, 4, 6, 1, 8, 1, 8, 8, 9, 9, 6, 6, 6, 7, 6, 7, 1, 3, 8, 7, 9, 5, 4, 8, 3, 4, 0, 7, 8, 1, 9, 4, 7, 3, 5, 0, 0, 2, 0, 8, 0, 9, 5 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS If A and B are neighboring vertices of a regular dodecahedron having center O, then the central angle AOB is this number; the exact value is arccos((1/3)*sqrt(5)) = arcsin(2/3). The (minimal) central angle of the other four regular polyhedra are as follows: - tetrahedron: A156546, - cube: A137914, - octahedron: A019669, - icosahedron: A105199. LINKS FORMULA The dodecahedron has 12 faces and 20 vertices. To find the central angle, we need any neighboring pair of vertices. Here are all 20 vertices: - (d,d,d) where d is 1 or -1 (that's 8 vertices); - (0, d*(t-1),d*t), where d is 1 or -1 and d = golden ratio = (1+sqrt(5))/2; - (d*(t-1), d*t, 0); and ((d*t,0,d*(t-1)). An example of a neighboring pair is (1,1,1) and (0,t,t-1). Apply the usual formula for the cosine of the angle between two vectors. EXAMPLE arccos((1/3)*sqrt(5))=0.729727656226966..., or, in degrees, 41.810314895778598065857916730578259531014119535901347753... MAPLE evalf(arcsin(2/3)); #  Robert FERREOL, Sep 14 2019 MATHEMATICA RealDigits[ArcCos[Sqrt[5]/3], 10, 120][[1]] (* Harvey P. Dale, Feb 23 2015 *) PROG (PARI) asin(2/3) \\ Charles R Greathouse IV, May 28 2013 CROSSREFS Sequence in context: A021936 A277525 A154176 * A180872 A003673 A021141 Adjacent sequences:  A156544 A156545 A156546 * A156548 A156549 A156550 KEYWORD nonn,cons AUTHOR Clark Kimberling, Feb 09 2009 STATUS approved

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Last modified August 5 21:58 EDT 2020. Contains 336213 sequences. (Running on oeis4.)